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Triple Integrals : Changing the Order of Integration and Finding the Volume of a Wedge

1. Using the integral ∫-1-->1 ∫x^2-->1 ∫0-->1-y dz dy dx
a) Sketch the region of integration.
Write the integral as an equivalent iterated integral in the order:
b) dy dz dx
c) dx dz dy
d) dz dx dy

2. Find the volume of a wedge cut from the cylinder x^2 +y^2 =1 by planes z=-y and z=0.
Please show me the process too.

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The solution is attached below (next to the paperclip icon) in two formats. one is in Word XP Format, while the other is in Adobe pdf format. Therefore you can choose the format that is most suitable to you.

The volume of integration can be seen in the figure below:

As you can see the z coordinate is bound between the plane z=0 on top and the line z=1-y on the top.
The y coordinate is bound by the plane y=1 from the right and the parabola y=x2 from the left.
x is simply bound between the planes x=±1 (front and back)

Conversion to dzdydx

We need to express y as functions of z and x, the limits of z as a function of x and the limits of x as constants.

Since the original limits of x are already constants, they stay the same.

The relation between y and z can be written as:

This is the upper limit of y. The original lower limit of y is x2. Since x is the last variable in the integration order, this can ...

Solution Summary

Triple Integrals, Changing the Order of Integration and Finding the Volume of a Wedge are investigated. The solution is detailed and well presented.

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